To emphasize the numerical challenge, we note that the EP matrix element \(M\)  requires the computation of the self-consistently screened potential due to the phonon displacements, then requiring evaluation of the matrix element between electron states \(k\)  and \(k^{\prime }\)  on the Fermi surface, with necessary band and phonon branch indices. This extensive computation has, through innovative algorithms, been brought to viable although still time-consuming level. The electron states are expressed in terms of localized Wannier functions,[1]} and the phonons are expressed in terms of localized lattice Wannier functions.[2]}, [3]} This combination considerably speeds the various required zone samplings. The linear response algorithms have been implemented to enable the phonon-induced change in screened potential to be calculated from the formal infinitesimal-displacement limit.
